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Nucleon Resonance Spectrum and Form Factors from Superconformal
Quantum Mechanics in Holographic QCD
Guy F. de Teramond
Universidad de Costa Rica
Nucleon Resonances:
From Photoproduction
to High Photon Virtualities
ECT*, Trento, 12 - 16 October 2015
In collaboration with Stan Brodsky and Hans G. Dosch
ECT*, N* Workshop, Trento, 15 October 2015Page 1
Quest for a semiclassical approximation to describe bound states in QCD
(Convenient starting point in QCD)
I. Semiclassical approximation to QCD in the light-front: Reduction of QCD LF Hamiltonian leads to a
relativistic LF wave equation, where complexities from strong interactions are incorporated in effective
potential U
II. Construction of LF potential U: Since the LF semiclassical approach leads to a one-dim QFT, it is
natural to extend conformal and superconformal QM to the light front since it gives important insights
into the confinement mechanism, the emergence of a mass scale and baryon-meson SUSY
III. Correspondence between equations of motion for arbitrary spin in AdS space and relativistic LF bound-
state equations in physical space-time: Embedding of LF wave equations in AdS leads to extension of
LF potential U to arbitrary spin from conformal symmetry breaking in the AdS5 action
ECT*, N* Workshop, Trento, 15 October 2015Page 2
Outline of this talk
1 Semiclassical approximation to QCD in the light front
2 Conformal quantum mechanics and light-front dynamics: Mesons
3 Embedding integer-spin wave equations in AdS space
4 Superconformal quantum mechanics and light-front dynamics: Baryons
5 Superconformal baryon-meson symmetry
6 Light-front holographic cluster decomposition and form factors
ECT*, N* Workshop, Trento, 15 October 2015Page 3
(1) Semiclassical approximation to QCD in the light front
• Start with SU(3)C QCD Lagrangian
LQCD = ψ (iγµDµ −m)ψ − 14G
aµνG
aµν
• Express the hadron four-momentum generator P = (P+, P−,P⊥) in terms of dynamical fields
ψ+ = Λ±ψ and A⊥(Λ± = γ0γ±
)quantized in null plane x+ = x0 + x3 = 0
P− = 12
∫dx−d2x⊥ψ+ γ
+ (i∇⊥)2 +m2
i∂+ψ+ + interactions
P+ =∫dx−d2x⊥ψ+γ
+i∂+ψ+
P⊥ = 12
∫dx−d2x⊥ψ+γ
+i∇⊥ψ+
• LF invariant Hamiltonian P 2 = PµPµ = P−P+ −P2
⊥
P 2|ψ(P )〉 = M2|ψ(P )〉
where |ψ(P )〉 is expanded in multi-particle Fock states |n〉: |ψ〉 =∑
n ψn|n〉
ECT*, N* Workshop, Trento, 15 October 2015Page 4
Effective QCD LF Bound-state Equation
[GdT and S. J. Brodsky, PRL 102, 081601 (2009)]
• Factor out the longitudinal X(x) and orbital kinematical dependence from LFWF ψ
ψ(x, ζ, ϕ) = eiLϕX(x)φ(ζ)√2πζ
• Ultra relativistic limit mq → 0 longitudinal modes X(x) decouple and LF Hamiltonian equation
PµPµ|ψ〉 = M2|ψ〉 is a LF wave equation for φ(
− d2
dζ2− 1− 4L2
4ζ2+ U(ζ)
)φ(ζ) = M2φ(ζ)
• Invariant transverse variable in impact space
ζ2 = x(1− x)b2⊥
conjugate to invariant massM2 = k2⊥/x(1− x)
• Critical valueL = 0 corresponds to lowest possible stable solution: ground state of the LF Hamiltonian
• Relativistic and frame-independent LF Schrodinger equation: U is instantaneous in LF time and com-
prises all interactions, including those with higher Fock states.
ECT*, N* Workshop, Trento, 15 October 2015Page 5
(2) Conformal quantum mechanics and light-front dynamics[S. J. Brodsky, GdT and H.G. Dosch, PLB 729, 3 (2014)]
• Incorporate in 1-dim effective QFT the conformal symmetry of 4-dim QCD Lagrangian in the limit of
massless quarks: Conformal QM [V. de Alfaro, S. Fubini and G. Furlan, Nuovo Cim. A 34, 569 (1976)]
• Conformal Hamiltonian:
H = 12
(p2 +
g
x2
)g dimensionless: Casimir operator of the representation
• Schrodinger picture: p = −i∂x
H = 12
(− d2
dx2+
g
x2
)• QM evolution
H|ψ(t)〉 = id
dt|ψ(t)〉
H is one of the generators of the conformal group Conf(R1). The two additional generators are:
• Dilatation: D = −14 (px+ xp)
• Special conformal transformations: K = 12x
2
ECT*, N* Workshop, Trento, 15 October 2015Page 6
• H , D and K close the conformal algebra
[H,D] = iH, [H,K] = 2iD, [K,D] = −iK
• dAFF construct a new generator G as a superposition of the 3 generators of Conf(R1)
G = uH + vD + wK
and introduce new time variable τ
dτ =dt
u+ vt+ wt2
• Find usual quantum mechanical evolution for time τ
G|ψ(τ)〉 = id
dτ|ψ(τ)〉 H|ψ(t)〉 = i
d
dt|ψ(t)〉
G =12u
(− d2
dx2+
g
x2
)+i
4v
(xd
dx+
d
dxx
)+
12wx2.
• Operator G is compact for 4uw − v2 > 0, but action remains conformal invariant !
• Emergence of scale: Since the generators of Conf(R1) ∼ SO(2, 1) have different dimensions a
scale appears in the new Hamiltonian G, which according to dAFF may play a fundamental role
ECT*, N* Workshop, Trento, 15 October 2015Page 7
Connection to light-front dynamics
• Compare the dAFF Hamiltonian G
G =12u
(− d2
dx2+
g
x2
)+i
4v
(xd
dx+
d
dxx
)+
12wx2.
with the LF Hamiltonian HLF
HLF = − d2
dζ2− 1− 4L2
4ζ2+ U(ζ)
and identify dAFF variable x with LF invariant variable ζ
• Choose u = 2, v = 0
• Casimir operator from LF kinematical constraints: g = L2 − 14
• w = 2λ2 fixes the LF potential to harmonic oscillator in the LF plane λ2 ζ2
U ∼ λ2ζ2
ECT*, N* Workshop, Trento, 15 October 2015Page 8
(3) Embedding integer spin wave equations in AdS space[GdT, H.G. Dosch and S. J. Brodsky, PRD 87, 075004 (2013)]
• Integer spin-J in AdS conveniently described by tensor field ΦN1···NJ with effective action
Seff =∫ddx dz
√|g| eϕ(z) gN1N ′1 · · · gNJN ′J
(gMM ′DMΦ∗N1...NJ
DM ′ΦN ′1...N′J
− µ2eff (z) Φ∗N1...NJ
ΦN ′1...N′J
)DM is the covariant derivative which includes affine connection and dilaton ϕ(z) effectively breaks
maximal symmetry of AdSd+1
ds2 =R2
z2
(dxµdx
µ − dz2)
• Effective mass µeff (z) is determined by precise mapping to light-front physics
• Non-trivial geometry of pure AdS encodes the kinematics and additional deformations of AdS encode
the dynamics, including confinement
ECT*, N* Workshop, Trento, 15 October 2015Page 9
• Physical hadron has plane-wave and polarization indices along 3+1 physical coordinates and a profile
wavefunction Φ(z) along holographic variable z
ΦP (x, z)µ1···µJ = eiP ·xΦ(z)µ1···µJ , Φzµ2···µJ = · · · = Φµ1µ2···z = 0
with four-momentum Pµ and invariant hadronic mass PµPµ=M2
• Variation of the action gives AdS wave equation for spin-J field Φ(z)ν1···νJ = ΦJ(z)εν1···νJ (P )[−z
d−1−2J
eϕ(z)∂z
(eϕ(z)zd−1−2J
∂z
)+(µR
z
)2]
ΦJ = M2ΦJ
with
(µR)2 = (µeff (z)R)2 − Jz ϕ′(z) + J(d− J + 1)
and the kinematical constraints to eliminate the lower spin states J − 1, J − 2, · · ·
ηµνPµ ενν2···νJ = 0, ηµν εµνν3···νJ = 0
• Kinematical constrains in the LF imply that µ must be a constant
[See also: T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, 076003 (2012)]
ECT*, N* Workshop, Trento, 15 October 2015Page 10
Light-front mapping[GdT and S. J. Brodsky, PRL 102, 081601 (2009)]
• Upon substitution ΦJ(z) ∼ z(d−1)/2−Je−ϕ(z)/2 φJ(z) and z→ζ in AdS WE[−z
d−1−2J
eϕ(z)∂z
(eϕ(z)
zd−1−2J∂z
)+(µR
z
)2]
ΦJ(z) = M2ΦJ(z)
we find LFWE (d = 4)(− d2
dζ2− 1− 4L2
4ζ2+ U(ζ)
)φJ(ζ) = M2φJ(ζ)
ctct
y
z
1-20118811A3
withU(ζ) = 1
2ϕ′′(ζ) +
14ϕ′(ζ)2 +
2J − 32z
ϕ′(ζ)
and (µR)2 = −(2− J)2 + L2
• Unmodified AdS equations correspond to the kinetic energy terms for the partons
• Effective confining potential U(ζ) corresponds to the IR modification of AdS space
• AdS Breitenlohner-Freedman bound (µR)2 ≥ −4 equivalent to LF QM stability condition L2 ≥ 0
ECT*, N* Workshop, Trento, 15 October 2015Page 11
Meson spectrum
• Dilaton profile in the dual gravity model determined from one-dim QFTh (dAFF)
ϕ(z) = λz2, λ2 = 12w
• Effective potential: U = λ2ζ2 + 2λ(J − 1)
• LFWE (− d2
dζ2− 1− 4L2
4ζ2+ λ2ζ2 + 2λ(J − 1)
)φJ(ζ) = M2φJ(ζ)
• Normalized eigenfunctions 〈φ|φ〉 =∫dζ φ2(z) = 1
φn,L(ζ) = |λ|(1+L)/2
√2n!
(n+L)!ζ1/2+Le−|λ|ζ
2/2LLn(|λ|ζ2)
• Eigenvalues for λ > 0 M2n,J,L = 4λ
(n+
J + L
2
)• λ < 0 incompatible with LF constituent interpretation
ECT*, N* Workshop, Trento, 15 October 2015Page 12
Three relevant points . . .
• A linear potential Veff in the instant form implies a quadratic potential Ueff in the front form at large
distances→ Regge trajectories
Ueff = V 2eff + 2
√p2 +m2
q Veff + 2Veff
√p2 +m2
q
[A. P. Trawinski, S. D. Glazek, S. J. Brodsky, GdT, H. G. Dosch, PRD 90, 074017 (2014)]
• Results are easily extended to light quarks
[S. J. Brodsky, GdT, H. G. Dosch and J. Erlich, Phys. Rept. 584, 1 (2015)
∆M2mq ,mq =
∫ 10 dx e
− 1λ
(m2qx
+m2q
1−x
) (m2q
x +m2q
1−x
)∫ 1
0 dx e− 1λ
(m2qx
+m2q
1−x
)• For n partons invariant LF variable ζ is [S. J. Brodsky and GdT, PRL 96, 201601 (2006)]
ζ =√
x
1− x∣∣ n−1∑j=1
xjb⊥j∣∣
where xj and x are longitudinal momentum fractions of quark j in the cluster and of the active quark
ECT*, N* Workshop, Trento, 15 October 2015Page 13
0
1
2
3
4
5
0
(a)
1L
M2 (G
eV
2)
2 3
n=2 n=1 n=0 n=2 n=1 n=0
π(1800)
π(1880)
π2(1670)
b1(1235)
π(1300)
π(140)K(494)
K1(1270)
K1(1400)
K2(1820)
K2(1770)
0
(b)
1L
2 37-20148851A8
0 2 4
L
0
2
4
6(a)
M2 (G
eV
2)
n=3 n=2 n=1 n=0
ω(782)
ρ(770)
ω(1420)
ρ(1450)
ω(1650)
ρ(1700)
0 2 4
L
(b) n=2 n=1 n=0
K*(892)
K*2(1430)
K*3(1780)
K*4(2045)
K*(1410)
K*(1680)
7-20148851A9
f2(2300)
f2(1950)
a2(1320)
f2(1270)
a4(2040)
f4(2050)
ρ3(1690)
ω3(1670)
0 2 4
1
3
5
φ(1020)
φ(1680)
φ(2170)
n=3 n=2 n=1 n=0
φ3(1850)
L2-20158872A5
M2 (
Ge
V2)
Orbital and radial excitations for√λ = 0.59 GeV (pseudoscalar) and 0.54 GeV (vector mesons)
ECT*, N* Workshop, Trento, 15 October 2015Page 14
(4) Superconformal quantum mechanics and light-front dynamics[GdT, H.G. Dosch and S. J. Brodsky, PRD 91, 045040 (2015)]
• SUSY QM contains two fermionic generators Q and Q†, and a bosonic generator, the Hamiltonian H
[E. Witten, NPB 188, 513 (1981)]
• Closure under the graded algebra sl(1/1):
12{Q,Q
†} = H
{Q,Q} = {Q†, Q†} = 0
[Q,H] = [Q†, H] = 0
Note: Since [Q†, H] = 0 the states |E〉 and Q†|E〉 have identical eigenvalues E
• A simple realization is
Q = χ (ip+W ) , Q† = χ† (−ip+W )
where χ and χ† are spinor operators with anticommutation relation
{χ, χ†} = 1
• In a 2× 2 Pauli-spin matrix representation: χ = 12 (σ1 + iσ2) , χ† = 1
2 (σ1 − iσ2)
[χ, χ†] = σ3
ECT*, N* Workshop, Trento, 15 October 2015Page 15
• Following Fubini and Rabinovici consider a 1-dim QFT invariant under conformal and supersymmetric
transformations [S. Fubini and E. Rabinovici, NPB 245, 17 (1984)]
• Conformal superpotential (f is dimensionless )
W (x) =f
x
• Thus 1-dim QFT representation of the operators
Q = χ
(d
dx+f
x
), Q† = χ†
(− d
dx+f
x
)• Conformal Hamiltonian H = 1
2{Q,Q†} in matrix form
H = 12
− d2
dx2 + f(f−1)x2 0
0 − d2
dx2 + f(f+1)x2
• Conformal graded-Lie algebra has in addition to Hamiltonian H and supercharges Q and Q†, a new
operator S related to generator of conformal transformations K ∼ {S, S†}
S = χx, S† = χ†x
ECT*, N* Workshop, Trento, 15 October 2015Page 16
• Find enlarged algebra (Superconformal algebra of Haag, Lopuszanski and Sohnius (1974))
12{Q,Q
†} = H, 12{S, S
†} = K
12{Q,S
†} =f
2+σ3
4+ iD
12{Q
†, S} =f
2+σ3
4− iD
where the operators
H =12
(− d2
dx2+f2 − σ3f
x2
)D =
i
4
(d
dxx+ x
d
dx
)K = 1
2x2
satisfy the conformal algebra
[H,D] = iH, [H,K] = 2iD, [K,D] = −iK
ECT*, N* Workshop, Trento, 15 October 2015Page 17
• Following F&R define a supercharge R, a linear combination of the generators Q and S
R =√uQ+
√wS
and consider the new generator G= 12{R,R
†} which also closes under the graded algebra sl(1/1)
12{R,R
†} = G
{R,R} = {R†, R†} = 0
[R,H] = [R†, H] = 0
12{Q,Q
†} = H
{Q,Q} = {Q†, Q†} = 0
[Q,H] = [Q†, H] = 0
• New QM evolution operator
G = uH + wK + 12
√uw (2f + σ3)
is compact for uw > 0: Emergence of a scale since Q and S have different units
• Light-front extension of superconformal results follows from
x→ ζ, f → ν + 12 , σ3 → γ5, 2G→ HLF
• Obtain:
HLF = − d2
dζ2+
(ν + 1
2
)2ζ2
−ν + 1
2
ζ2γ5 + λ2ζ2 + λ(2ν + 1) + λγ5
where coefficients u and w are fixed to u = 2 and w = 2λ2
ECT*, N* Workshop, Trento, 15 October 2015Page 18
Nucleon Spectrum
• In 2× 2 block-matrix form
HLF =
− d2
dζ2− 1−4ν2
4ζ2+ λ2ζ2 + 2λ(ν + 1) 0
0 − d2
dζ2− 1−4(ν+1)2
4ζ2+ λ2ζ2 + 2λν
• Eigenfunctions
ψ+(ζ) ∼ ζ12
+νe−λζ2/2Lνn(λζ2)
ψ−(ζ) ∼ ζ32
+νe−λζ2/2Lν+1
n (λζ2)
• Eigenvalues
M2 = 4λ(n+ ν + 1)
• Lowest possible state n = 0 and ν = 0
• Orbital excitations ν = 0, 1, 2 · · · = L
• L is the relative LF angular momentum
between the active quark and spectator clusterL
n � 0n = 1n � 2n � 3
NH940L
NH1440L
NH1710L NH1720LNH1680L
NH1900L
NH2220L
Λ � 0.49 GeV
M2IGeV2M
0 1 2 3 40
1
2
3
4
5
6
7
ECT*, N* Workshop, Trento, 15 October 2015Page 19
(6) Superconformal baryon-meson symmetry[H.G. Dosch, GdT, and S. J. Brodsky, PRD 91, 085016 (2015)]
• Previous application: positive and negative chirality components of baryons related by superchargeR
R†|ψ+〉 = |ψ−〉
with identical eigenvalue M2 since [R,G] = [R†, G] = 0
• Conventionally supersymmetry relates fermions and bosons
R|Baryon〉 = |Meson〉 or R† |Meson〉 = |Baryon〉
• If |φ〉M is a meson state with eigenvalueM2,G |φ〉M = M2|φ〉M , then there exists also a baryonic
state R† |φ〉M = |φ〉B with the same eigenvalue M2:
G |φ〉B = GR†|φ〉M = R†G |φ〉M = M2|φ〉B
• For a zero eigenvalue M2 we can have the trivial solution
|φ(M2 = 0)〉B = 0
Special role played by the pion as a unique state of zero energy
ECT*, N* Workshop, Trento, 15 October 2015Page 20
Baryon as superpartner of the meson trajectory |φ〉 =
φMeson
φBaryon
• Compare superconformal meson-baryon equations with LFWE for nucleon (leading twist) and pion:
(− d2
dx2+ λ2 x2 + 2λ f + λ+
4(f − 12)2 − 1
4x2
)φBaryon = M2φBaryon
(− d2
dζ2+ λ2
B ζ2 + 2λB(LB + 1) +
4L2B − 14ζ2
)ψ+LB
= M2 ψ+LB(
− d2
dx2+ λ2 x2 + 2λ f − λ+
4(f + 12)2 − 1
4x2
)φMeson = M2φMeson
(− d2
dζ2+ λ2
M ζ2 + 2λM (LM − 1) +4L2
M − 14ζ2
)φLM = M2 φLM
• Find: λ = λM = λB , f = LB + 12 = LM − 1
2 ⇒ LM = LB + 1
ECT*, N* Workshop, Trento, 15 October 2015Page 21
b1
π2
0
Nπ
2 4
0
2
4
6
LM = LB + 11-20158872A1
M2 (
Ge
V2)
1–2
+
N1–2 N
3–2
- -
N3–2 N
5–2
+ +
N9–2
+
0
2
4
6
ρ,ω
a2,f2
ρ3,ω3
a4,f4
0 2 4
LM = LB + 11-20158872A3
M2 (
Ge
V2)
Δ3–2
+
Δ1–2
-
,Δ3–2
-
Δ1–2
+
Δ11–2
+
,Δ3–2
+
,Δ5–2
+
,Δ7–2
+
Superconformal meson-nucleon partners: solid line corresponds to√λ = 0.53 GeV
ECT*, N* Workshop, Trento, 15 October 2015Page 22
Supersymmetry across the light and heavy-light hadronic spectrum[H.G. Dosch, GdT, and S. J. Brodsky, Phys. Rev. D 92, 074010 (2015)]
• Introduction of quark masses breaks conformal symmetry without violating supersymmetry
0 1
LM = LB + 1
Ds1 (2536)
M2 (G
eV
2)
2
4
6
8
0
Ds1 (2460)
Ds,
[I ] c,
[I ] c
[I ] c
[I ] c
[I ] c
1
LM = LB + 1
D*s2 (2573)
D*s,
[I ] *c
Ds
D*s
D1 (2420)
M2 (G
eV
2)
2
4
6
8
Σc
D*2 (2460)
Σc (2520)
D,Λc,Σc D*,Σ*
DD* (2007)
Λc
(2645)
4-20158880A1
(a) (b)
(c) (d)
Supersymmetric relations between mesons and baryons with charm
ECT*, N* Workshop, Trento, 15 October 2015Page 23
LM = LB + 1LM = LB + 1
Bs,
[I ] b
[I ]
[I ]
b
B*s,
[I ] *b
Bs
M2 (G
eV
2)
M2 (G
eV
2)
B,Λb,Σb
Σb
0 1
26
30
34
38
26
30
34
38
b(5945)
B*s
0 1
B*,Σb*
B B*
B1 (5721)
Bs1 (5830)
Λb
Σb*
B*2 (5747)
B*s2 (5840)
4-20158880A2
(a) (b)
(c) (d)
Supersymmetric relations between mesons and baryons with beauty
ECT*, N* Workshop, Trento, 15 October 2015Page 24
Emerging SUSY from color dynamics 3→ 3× 3 Work in progress[S. J. Brodsky, GdT, H. G. Dosch, C. Lorce]
• Superconformal spin-dependent Hamiltonian to describe mesons and baryons (chiral limit)
G = {R†λ, Rλ}+ 2λ I s R ∼ Q+ λS
• LFWE for mesons(− d2
dζ2−
1− 4L2M
4ζ2+ λ2ζ2 + 2λ(LM + s− 1)
)φMeson = M2φMeson
• LFWE for nucleons(− d2
dζ2−
1− 4L2B
4ζ2+ λ2 ζ2 + 2λ(LB + s+ 1)
)φBaryon = M2 φBaryon
with LM = LB + 1
• Spin of the spectator cluster s is the spin of the corresponding meson !
ECT*, N* Workshop, Trento, 15 October 2015Page 25
Preliminary
ECT*, N* Workshop, Trento, 15 October 2015Page 26
• How good is semiclassical approximation based on superconformal QM and LF clustering properties?
Best fit for the hadronic scale√λ from the different sectors including radial and orbital excitations
Preliminary
ECT*, N* Workshop, Trento, 15 October 2015Page 27
(6) Light-front holographic cluster decomposition and form factors[S. J. Brodsky, GdT, H. G. Dosch, C. Lorce] Work in progress
• LF Holographic FF Fτ=N (Q2) expressed as the N − 1 product of poles for twist τ = NS. J. Brodsky and GdT, PRD 77, 056007 (2008)
Fτ=2(Q2) =1(
1 + Q2
M2ρ
)Fτ=3(Q2) =
1(1 + Q2
M2ρ
)(1 + Q2
M2ρ′
)· · ·
Fτ=N (Q2) =1(
1 + Q2
M2ρ
)(1 + Q2
M2ρ′
)· · ·(
1 + Q2
M2ρN−2
)• Spectral formula
M2ρn → 4κ2 (n+ 1/2)
• Cluster decomposition in terms of twist τ = 2 FFs !
Fτ=N (Q2) = Fτ=2
(Q2)Fτ=2
(13Q2
)· · ·Fτ=2
(1
2N − 3Q2
)
ECT*, N* Workshop, Trento, 15 October 2015Page 28
• Example: Dirac proton FF F p1in terms of the pion form factor Fπ :
F p1 (Q2) = Fπ(Q2)Fπ
(13Q2
)(equivalent to τ = 3 FF)
0
0.4
0.8
1.2
10 20 300
Q2 (GeV2)
Q4
Fp 1
(Q
2)
(G
eV
4)
2-20128820A18
• But . . . we know that higher Fock components are required.
Example time-like pion FF:
|π〉 = ψqq/π|qq〉τ=2 + ψqqqq|qqqq〉τ=4 + · · ·
Fπ(q2) = (1− γ)Fτ=2(q2) + γFτ=4(q2)
Pqqqq = 12.5%
S. J. Brodsky, GdT, H. G. Dosch and J. Erlich, PR 584, 1 (2015)
2
0
0
s (GeV2)lo
g Fπ(s
)
-2 2
-2
7-20148851A6
ECT*, N* Workshop, Trento, 15 October 2015Page 29
• Transition form factors for the radial transition n = 0→ n = 1:
Fn=0→1τ=2 (Q2) =
12
Q2
M2ρ(
1 + Q2
M2ρ
)(1 + Q2
M2ρ′
)Fn=0→1τ=3 (Q2) =
√2
3
Q2
M2ρ(
1 + Q2
M2ρ
)(1 + Q2
M2ρ′
)(1 + Q2
M2ρ′′
)· · ·
Fn=0→1τ=N (Q2) =
√N − 1N
Q2
M2ρ(
1 + Q2
M2ρ
)(1 + Q2
M2ρ′
)· · ·(
1 + Q2
M2ρN−1
)where Fn=0→1
τ=N (Q2) is expressed as the N product of poles
• LF cluster decomposition: Express the transition form factor as the product of the pion transition form
factor times the N − 1 product of pion elastic form factors evaluated at different scales
Fn=0→1τ=N (Q2) =
√N − 1N
Fn=0→1τ=2
(Q2)Fτ=2
(13Q2
)· · ·Fτ=2
(1
2(N + 1)− 3Q2
).
(1)
ECT*, N* Workshop, Trento, 15 October 2015Page 30
• Example: Dirac transition form factor of the proton
to a Roper state F p1N→N∗ :
F p1N→N∗(Q2) =
2√
23Fπ→π′(Q2)Fπ
(15Q2
)(equivalent to τ = 3 TFF )
[GdT and S. J. Brodsky, AIP Conf. Proc. 1432, 168 (2012)]
Old JLab data Q2 GeV2
F1p
N®N*IQ2M
0 1 2 3 4 50.00
0.05
0.10
0.15
0.20
• Holographic QCD computation including Ap1/2 and Sp1/2 :
T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, PRD 87, 016017 (2013)
• Data confirmed by recent JLab data. Possible solution to describe small (Q2 < 1 GeV2) data:
Include τ = 5 higher Fock component |qqqqq〉 in addition to τ = 3 valence |qqq〉
Fτ=3 ∼1Q4
, Fτ=5 ∼1Q8
ECT*, N* Workshop, Trento, 15 October 2015Page 31
Thanks !
For a review: S. J. Brodsky, GdT, H. G. Dosch and J. Erlich, Phys. Rept. 584, 1 (2015)
ECT*, N* Workshop, Trento, 15 October 2015Page 32